Fourier restriction theorems, whose study had been initiated by E.M. Stein, usually describe a family of a priori estimates of the -norm of the restriction of the Fourier transform of a function in to a given subvariety , endowed with a suitable measure. Such estimates allow to define the restriction of the Fourier transform of an -function to in an operator theoretic sense. In this article, we begin to investigate the question what is the „intrinsic" pointwise relation between and the Fourier transform of , by looking at curves in the plane, for instance with non-vanishing curvature. To this end, we bound suitable maximal operators, including the Hardy–Littlewood maximal function of the Fourier transform of restricted to .
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Detlef Müller, Fulvio Ricci, James Wright, A maximal restriction theorem and Lebesgue points of functions in . Rev. Mat. Iberoam. 35 (2019), no. 3, pp. 693–702DOI 10.4171/RMI/1066